BOOK V

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S 47. The fifth book of the _Elements_ is not exclusively geometrical. It contains the theory of ratios and proportion of quantities in general. The treatment, as here given, is admirable, and in every respect superior to the algebraical method by which Euclid's theory is now generally replaced. We shall treat the subject in order to show why the usual algebraical treatment of proportion is not really sound. We begin by quoting those definitions at the beginning of